Last week, House Majority Leader Eric Cantor of Virginia also promised to hold a vote on the amendment.

“Two weeks ago, leadership rolled their eyes and almost scoffed at the notion of the balanced budget amendment,” said freshman Republican Rep. Joe Walsh of Illinois in a recent conference call. “Leadership is taking this concept more seriously. Each and every day we’re getting more positive signals from leadership,” said Walsh.

“Do they love the idea? Probably not,” the congressman added. “But they’re getting more resigned to the fact that without this […] there are going to be a number of us who simply will not vote to raise the debt ceiling.”

But while all Senate Republicans joined in support of McConnell’s announcement Wednesday, some are taking McConnell’s extended olive branch with a level of frustration — that the party is playing of a weak card when stronger options should be on the table going into negotiations with Democrats.

One GOP source on Capitol Hill told The Daily Caller there’s been a push among the conservative wing of the Republican caucus for McConnell to tie the balanced budget amendment to a vote on the debt ceiling.

“There have been attempts to communicate that with leadership, certainly at the staff level,” the sources said. “That would be a very strong hand to play.” But, he said, “there is no appetite to do that in the leadership.”

“We have asked the leadership, ‘Okay, what hand do you want to play? Why not play the strongest hand and tie the balanced budget amendment to the debt ceiling?’ the source told TheDC. “The response has been unsatisfactory.”

Walsh, who sponsored the amendment in the House, also signed the “Cut, Cap and Balance” pledge vowing not to vote to raise the debt limit unless spending is cut and capped and the Constitution is amended to require a balanced budget.

As the August 2 deadline looms nearer, the strategy of the more conservative wing of Republican members on the Hill is clear, though it’s anyone’s guess as to whether it can be accomplished.

More moderate, rank-and-file Republicans still need to be convinced to sign the pledge and moderate Democrats still need to be won over. According to proponents of the pledge, the balanced budget amendment is crucial, as it is the only way to tell the public that lawmakers are working on a long-term solution and not just a short-term deal.

“There’s some trepidation,” said Walsh, when asked about his conversations with Republican colleagues. “[S]igning this pledge – it’s almost a blood oath.”

But while the pledge continues to make its way through the halls of Congress, there is a general uneasiness among the freshman lawmakers that their representatives in the debt limit talks — Boehner and McConnell — will not negotiate an acceptable deal.

“The Democrats are digging up everything to throw in the negotiating box,” the GOP source told TheDC. “They are walking in with a huge chunk of demands, and we’re not going to have any.”

In the conference call, Walsh voiced similar concerns, and said he is especially worried about last-minute maneuvers.

“I’m not optimistic about what’s being crafted right now behind the scenes,” said the congressman. “Even though I think the leadership understands how important permanent change is, their heart is still not into it, and they are probably going to craft a deal that certain members will not be happy with.”

“The fear is we’re not going to know what [the leadership] is negotiating for,” said Hill GOP source. “We need to know what the ask is. We need something very strong going into negotiations and the process needs to be transparent.”